Mathematical Physics

Yvonne Choquet-Bruhat

She applied fundamental mathematical results to provide
a firm basis for the solutions of
the classical field equations of physics, most
importantly those of general relativity and other
theories of gravitation, supergravity, and the non-Abelian
gauge theories of the standard model.

This is the author's (French) thesis, a magisterial work.
It exhaustively discusses the Cauchy problem for a system
of second order partial differential equations, linear in
second derivatives, having special relevance to the Einstein
equations in general relativity. The list of
results is lengthy and includes deep result
on exterior solutions and their uniqueness, on propagation
velocity of gravitational excitations, etc.

The first careful study of existence
theorems for non-analytic solutions of the Einstein
equations with various types of matter including
Kaluza-Klein unified, 5-dimensional,
extensions. It established that these general cases
define hyperbolic systems with well-defined Cauchy
problem.

A general method is presented that enable
one to construct asymptotic and approximate wave
solutions about a given solution for nonlinear system
of equations; this extends important earlier work, and
also shows when the Cauchy problem becomes ill-posed.
Applications are made to gravity.

The problem of existence and uniqueness of
global solutions of the constraint equations of general
relativity is studied in the important case of a closed
manifold, using general elliptic equation methods. The
essential results are that existence depends on delicate
properties of the manifold and on the sources of the
metric; the various cases are carefully classified.

This is perhaps the first study by a mathematician
of supergravity, the generalization of Einstein theory
unified, by a Grassmannian gauge invariance, with a massles
spin 3/2 fermionic field. In particular, the author extends
to supergravity the classic causality theorems that hold in
the purely geometric bosonic theory. The results are
extended both to N>1 supergravitie
and to higher dimensions, in particular to the currently
important maximal D=11 model.